Question

# There are 14 ice cream shops in a town, and they have the following property: Ice...

There are 14 ice cream shops in a town, and they have the following property: Ice cream shop number j sells j flavors of ice cream (Shop number 1 sells 1 flavor, shop number 2 sells 2 flavors... shop number 14 sells 14 flavors). A client randomly chooses an ice cream shop, and once inside, buys a scoop in each of the flavors available.

As part of a special offer, the client receives a lottery scratch card for each scoop of ice cream bought. On each scratch card, there is one number from the set {1, 2, . . . , 21}. Calculate the probability that the client entered ice cream shop number 2, if the number on all of the scratch cards was 1.

P( A | B ) = P (A and B) / P(B)

P( Shop = 2 | all scratch card =1 ) = P ( Shop = 2 and all scratch card =1) / P (all scratch card =1)

First we calculate,

P (all scratch card =1) = P( all scratch card =1 | Shop = 1) * P(Shop=1) + P( all scratch card =1 | Shop = 2) * P(Shop=2) ......+ P( all scratch card =1 | Shop = 14) * P(Shop=14)

Basically, this is a equation of calculating total probability,

P (all scratch card =1) = 1/21 * 1/14 + 1/21*1/21*1/14 + 1/21*1/21*1/21*1/14......+ (1/21)^14*1/14

= (1/21 * 1/14* ( 1 - (1/21)^14 ) )/ (1-1/21) = 0.00357142857

P ( Shop = 2 and all scratch card =1) = 1/14 * 1/21 * 1/21 = 0.00016196954

Thus,

P( Shop = 2 | all scratch card =1 ) = 0.00016196954/ 0.00357142857 = 0.0453514